Method and apparatus of estimating pure spectra and a concentration of a mixture

ABSTRACT

A method of estimating a pure spectrum and a concentration of each component constituting n sample mixtures, in which p kinds of components are mixed, includes (a) performing a principal component analysis of the spectra of the n mixtures, which are measured using m wavelengths, to represent the spectra of the n mixtures as factors and scores of the respective factors, wherein n, p, and m are integers and m&gt;p, and (b) performing an independent component analysis of the scores obtained in (a) to estimate the pure spectra and the concentrations of the respective components.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a spectral analysis of mixtures.More particularly, the present invention relates to a method andapparatus of estimating a pure spectrum and a concentration of eachcomponent constituting a mixture by performing a principal componentanalysis and an independent component analysis of the spectrum of themixture.

[0003] 2. Description of the Related Art

[0004] A principal component regression (PCR) is generally used toestimate a concentration of a spectrum of a mixture. PCR is amultivariate analysis including two steps. The first step is a principalcomponent analysis (PCA), in which the measured spectrum of a mixture isdecomposed into a product of a factor and a score using singular valuedecomposition (SVD). Typically, because the factor and the score,obtained by the SVD, do not exactly match the pure spectrum and theconcentration, it is difficult to estimate the pure spectra and theconcentration from the spectrum of the mixture using only PCA. For thisreason, to estimate the concentration, PCR requires information inaddition to the spectrum of the mixture, i.e., information on theconcentration of the mixture. In the second step, the additionalinformation, i.e., the concentration of the mixture, is regressed intothe score produced by the PCA to obtain a regression vector. Theregression vector, obtained from a regression equation of the score andthe concentration, is a contravariant vector of pure spectra ofcomponents other than a particular component to be estimated in themixture. However, the regression vector does not exactly match the purespectrum of the particular component. Although the regression vectorobtained by PCR enables estimation of the concentration of a particularcomponent from the spectrum of a mixture, it is still difficult toestimate the pure spectrum of the particular component.

[0005] Consequently, it is impossible to estimate the pure spectrum ofeach component contained in a mixture using PCR. In addition, toestimate the concentration of each component, PCR requires an additionalcalibration set including accurate information on the concentration ofthe mixture. Further, in a case where only the information on thespectrum of a mixture is given because calibration is actuallyimpossible, the multivariate analysis cannot be employed.

SUMMARY OF THE INVENTION

[0006] The present invention provides a method and apparatus ofestimating a pure spectrum and a concentration of each component of amixture using only the spectrum of the mixture by applying a principalcomponent analysis (PCA) and an independent component analysis (ICA) tothe spectrum of the mixture.

[0007] In accordance with an embodiment of the present invention, amethod of estimating a pure spectrum and a concentration of eachcomponent constituting n sample mixtures in which p kinds of componentsare mixed includes (a) performing a principal component analysis of thespectra of the n mixtures, which are measured using m wavelengths, torepresent the spectra of the n mixtures as factors and scores of therespective factors, wherein n, p, and m are integers and m>p, and (b)performing an independent component analysis of the scores obtained in(a) to estimate the pure spectra and the concentrations of therespective components.

[0008] Preferably, the number of factors to be used is decided fromamong the factors obtained in (a) and the independent component analysisis applied to the scores of the decided factors. The concentrations ofthe components constituting the mixture may be statisticallyindependent.

[0009] Performing the independent component analysis may include (b1)performing the independent component analysis of the scores of thefactors to decompose the scores into a mixing matrix and independentcomponents, (b2) estimating the product of the factors obtained in (a)and the mixing matrix obtained in (b1) as the pure spectra of therespective components, and (b3) estimating the independent componentsobtained in step (b1) as being proportional to the concentrations of thecomponents contained in the mixture.

[0010] Performing the independent component analysis may include (b1)(b1) deciding the number of the factors to be used from among thefactors obtained in (a), (b2) performing the independent componentanalysis of the scores of the decided factors to decompose the scoresinto a mixing matrix and independent components, (b3) estimating theproduct of the decided factor and the mixing matrix obtained in (b2) asthe pure spectrum of each component, and (b4) estimating the independentcomponents obtained in (b2) as being proportional to the concentrationsof the components contained in the mixture.

[0011] In accordance with another embodiment of the present invention,there is provided a computer-readable medium having embodied thereon afirst program for performing a principal component analysis of a spectraof the n mixtures measured using m wavelengths to represent the spectraas factors and scores of the respective factors, wherein n and m areintegers, and a second program for performing an independent componentanalysis of the scores produced by the first program to decompose thescores into a mixing matrix and independent components, estimating thatthe product of the factor obtained by the first program and the mixingmatrix is the pure spectra of each component, and estimating that theindependent components are proportional to the concentrations of thecomponents contained in the mixture.

[0012] In accordance with still another embodiment of the presentinvention, an apparatus of estimating a pure spectrum and aconcentration of each component constituting n sample mixtures, in whichp kinds of components are mixed includes a principal component analysisunit for performing a principal component analysis of the spectra of then mixtures, which are measured using m wavelengths, to represent thespectra of the n mixtures as factors and scores of the respectivefactors, where n, m, and p are integers and m>p, an independentcomponent analysis unit for performing an independent component analysisof the scores provided from the principal component analysis unit, todecompose the scores into a mixing matrix and independent components, apure spectrum estimating unit for estimating the product of the factorprovided from the principal component analysis unit and the mixingmatrix as the pure spectra of each component, and a concentrationestimating unit for estimating the independent components as theconcentrations of the components contained in the mixture.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The above and other features and advantages of the presentinvention will become more apparent to those of ordinary skill in theart by describing in detail preferred embodiments thereof with referenceto the attached drawings in which:

[0014]FIG. 1 is a flowchart illustrating a method of estimating purespectra and concentrations of each component constituting a mixtureaccording to a preferred embodiment of the present invention;

[0015]FIGS. 2A through 2C are graphs showing a relationship of eachfactor and a press for explaining step 13 as shown in FIG. 1;

[0016]FIG. 3 shows pure spectra of sucrose and glucose;

[0017]FIG. 4 shows spectra of twenty-five (25) mixtures produced bymixing sucrose and glucose with different concentrations;

[0018]FIGS. 5A and 5B show pure spectra of sucrose and glucose obtainedby applying the present invention to the spectra of the mixtures shownin FIG. 4;

[0019]FIGS. 6A and 6B show concentrations of sucrose and glucoseobtained by applying the present invention to the spectra of themixtures shown in FIG. 4; and

[0020]FIG. 7 is a block diagram of an apparatus of estimating purespectra and concentrations of a mixture according to a first embodimentof the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0021] Korean Patent Application No. 2003-5198, filed on Jan. 27, 2003,and entitled: “Method and Apparatus of Estimating Pure Spectra and aConcentration of a Mixture,” is incorporated by reference herein in itsentirety.

[0022] The present invention will now be described more fullyhereinafter with reference to the accompanying drawings, in whichpreferred embodiments of the invention are shown. The invention may,however, be embodied in different forms and should not be construed aslimited to the embodiments set forth herein. Rather, these embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the invention to those skilled in theart. Like reference numerals refer to like elements throughout.

[0023] A spectrum of a mixture may be expressed as a linear combinationof the spectra of components contained in the mixture as shown inEquation (1):

A _(w) =A _(a,w) +A _(b,w) +A _(c,w)  (1)

[0024] where A_(w) represents a spectrum of a mixture measured using awavelength w, and A_(a,w), A_(b,w), and A_(c,w) represent spectra ofcomponents a, b, and c, respectively, measured using a wavelength w.

[0025] Meanwhile, a spectrum of each component may be expressed as theproduct of a pure spectrum and concentration as shown in Equation (2):

A _(c,w) =K _(c,w) ×C _(c)  (2)

[0026] where A_(c,w) represents the spectrum of component c measuredusing a wavelength w, K_(c,w) represents the pure spectrum of thecomponent c, measured using a wavelength w, and C_(c) represents theconcentration of the component c.

[0027] From Equations (1) and (2), the spectrum of the mixture can beexpressed as Equation (3):

A _(w) =K _(a,w) ×C _(a) +K _(b,w) ×C _(b) +K _(c,w) ×C _(c)  (3)

[0028] In addition, in a case where several mixtures are produced bymixing respective components with different concentrations, the spectrumof one of the mixtures can be expressed as Equation (4):

A _(w,n) =K _(a,w) ×C _(a,n) +K _(b,w) ×C _(b,n) +K _(c,w) ×C_(c,n)  (4)

[0029] where A_(w,n) represents the spectrum of the nth mixture measuredusing a wavelength w, K_(a,w), K_(b,w), and K_(c,w) represent the purespectra of components a, b, and c, respectively, measured using awavelength w, and C_(a,n), C_(b,n), and C_(c,n) represent theconcentrations of the components a, b, and c, respectively, of then^(th) mixture.

[0030] Equation (4) can be expressed in a matrix form as shown inEquation (5):

A=M·C  (5)

[0031] That is, the spectrum A of a mixture can be expressed by theproducts of the pure spectra M and the concentrations C of thecomponents thereof. Here, when the number of mixtures is n, the numberof wavelengths at which the spectra of mixtures are measured is m, andthe number of components is p, where n, m, and p are integers, and m>p,the dimension of matrix A is (m,n), the dimension of matrix B is (m,n),and the dimension of matrix C is (p,n).

[0032] Hereinafter, the present invention will be described in detailbased on spectral characteristics of mixtures.

[0033]FIG. 1 is a flowchart illustrating a method of estimating the purespectrum and the concentration of each component from the spectrum of amixture according to a preferred embodiment of the present invention.Preferably, the method includes, in step 11, performing a principalcomponent analysis (PCA), in step 13, deciding the number of factors, instep 15, performing an independent component analysis (ICA), and, instep 17, estimating the pure spectra and concentration.

[0034] Referring to FIG. 1, in step 11, the measured spectrum A of themixture is analyzed using the PCA and is decomposed into basic variates.For this, in step 11, singular value decomposition (SVD) is applied tothe measured spectrum A of the mixture and decomposes the spectrum Ainto the product of a factor F and a score S as shown in Equation (6):

A=USV′=FS  (6)

[0035] where the factor F is the common variate of the spectrum A of themixture and denotes an eigenvector or a principal component. The score Sdenotes a scaling coefficient corresponding to each principal component.

[0036] Supposing that the number of the mixtures is n, the number of thewavelengths at which the spectrum of the mixture is measured is m, andthe number of the components is p, the dimension of matrix A is (m,n), Uis an orthogonal matrix of dimension (m,m), V is an orthogonal matrix ofdimension (n,n), and S is a diagonal matrix of dimension (m,n), whichconsists of singular values whose covariance σ_(ij) (where i≠j) is zero(0). Meanwhile, the dimension of the factor F is (m,p), while thedimension of the score S is (p,n). Referring to Equation (6), it may beseen that the factor F is US, and the score S is V′. That is, accordingto the PCA, principal components occupying a large portion (e.g., 80% to90%) of total variance of original variables are decided as optimumfactors to be used in the next step. Subsequently, other componentsoccupying a smaller portion may be regarded as noise and removed. Thus,the score obtained by the PCA no longer has correlation.

[0037] Meanwhile, in a case where the SVD is applied to the purespectrum as shown in Equation (5), we suppose that the pure spectrum Mcan be expressed as in Equation (7):

M=usv′  (7)

[0038] where, when the concentrations of the components contained in themixture are statistically independent, because a covariance matrix ofthe concentration C as shown in Equation (5) becomes a unit matrix,Equation (8) is obtained based on Equations (6) and (7) as follows:

U=u, S=s, V′=v′C  (8)

[0039] Accordingly, the factor F, the score S, the pure spectrum M, andthe concentration C are in a relationship as shown in Equation (9):

M=usv′=USv′=Fv′

C=(v′)⁻¹ V′=(v′)⁻¹ S

S=v′C  (9)

[0040] More specifically, the pure spectrum M is obtained by multiplyingthe factor F by matrix v′, and the concentration C is obtained from thescore S multiplied by (v′)⁻¹.

[0041] Meanwhile, before performing the PCA in step 11, a low-noise bandincluding information on a component to be estimated among the spectraof the n mixtures measured using the m wavelengths is decided as ananalysis band. For this, the spectrum is recomposed from the score S andthe factor F, which are calculated by the PCA. Next, a difference (i.e.,a residual) between the recomposed spectrum and the initial spectrum Ais calculated. Thus, a range having a larger residual is normallydecided as a high-noise range. In addition, before performing the PCA instep 11, the spectra of the n mixtures undergo data preprocessing, suchas multiplicative scatter correction (MSC), mean-centering, andautoscaling, in order to remove scattering and noise contained in thespectra of the human tissue.

[0042] In step 13, the number of factors to be used for performingindependent component analysis in step 15 is decided from among thefactors F produced by the PCA in step 11. Step 13 will be described indetail with reference to FIGS. 2A through 2C. FIGS. 2A and 2B are graphsshowing a relationship between each factor and a press when there is nonoise. FIG. 2C is a graph showing a relationship between each factor andthe press when there is a noise.

[0043] In step 13, a statistic called “press” of each factor F producedby step 11 is calculated. If there is no noise, it is possible toestimate the number of components constituting a mixture using only thepress. That is, in a case where there is no noise, as shown in FIGS. 2Aand 2B, the numbers of factors F that have a non-zero press correspondsto the number of principal components contained in a mixture. Thus, itmay be seen that the mixtures contain two (2) and three (3) components,respectively.

[0044] However, since there generally is noise, as shown in FIG. 2C, thepress does not exactly have the value zero (0). In this case, a positionwhere the press is taken to be as zero (0) is statistically estimatedusing an F-test. Specifically, to perform the F-test, an f-ratio isobtained using Equation (10): $\begin{matrix}{F_{k} = {\frac{{PRESS}_{k} - {PRESS}_{\min}}{{PRESS}_{\min}} \cdot \frac{n}{n - k}}} & (10)\end{matrix}$

[0045] where F_(k) is the f-ratio of a k^(th) factor, PRESS_(k) is thepress value of the k^(th) factor, PRESS_(min) is the minimum pressvalue, and n is the number of the measured spectra.

[0046] Next, the f-value of each factor F is obtained using the f-ratioas a parameter. Finally, it is estimated that the number of the factors,of which f-values are more than 0.95, is equal to the number ofcomponents contained in a mixture. Here, the value “0.95” is selectedconsidering that the general estimation reliability is 95%.

[0047] In step 15, i.e., performing the ICA to obtain the matrix v′using Equation (9), even statistics of third or higher order, which arenot used in the step of performing the PCA, are made to be statisticallyindependent. More specifically, in a case where the concentrations C arestatistically independent at a higher order, non-zero values arepositioned only on the diagonal in a statistical multidimensional tensorof the concentrations C, while zero (0) is positioned at other portions.Meanwhile, as the score S obtained by the PCA equals v′C, even if v′ isorthogonal, non-zero values may be positioned on portions other than thediagonal in a multidimensional tensor of the v′C unlike that of theconcentrations C. Thus, a multidimensional tensor is obtained byrotating the score S to find a rotation v′ by which the value zero (0)is positioned only on the diagonal.

[0048] In step 15, the ICA is applied to the score S obtained in step11, preferably, the score S of the factor F decided in step 13, therebydecomposing the score S into a mixing matrix W and independentcomponents (ICs) as shown in Equation (11):

S=W·IC  (11)

[0049] wherein it is supposed that the number of mixtures is n, thenumber of components is p, the dimension of the score S is (p,n), andthe dimension of the independent components is (p,n).

[0050] In step 15, the ICA may be performed using maximization ofnon-gaussianity (MN), maximum likelihood estimation (MLE), minimizationof mutual information (MMI), or the like. Specifically, the algorithm“fastICA” of Hyvärinen may be used.

[0051] Meanwhile, both Equation (9) and Equation (11) express the scoreS decomposed into two matrixes. Also, though IC and C are expressed indifferent forms, the ICA is performed on the assumption that if IC and Chave the same statistics, IC is identical to C. Here, the statisticincludes not only second-order statistics used in the PCA but alsohigher-order statistics used in the ICA. Therefore, even if thesecond-order statistics of the score S obtained in the PCA are equal tothe second-order statistics of C, it cannot be inferred that the twoquantities are equal. However, if the higher-order statistics of thescore S obtained in the PCA is equal to the higher-order statistics ofIC obtained in the ICA, the two quantities may then be decided as equal.The present invention is described on the assumption that theconcentrations C of respective components are statistically independent,and the independent components IC obtained in the ICA are alsoindependent. Therefore, the concentration C and the independentcomponent C may be commonly used to express physical quantities and areproportional to one another. That is, the concentration C can beexpressed as C=proportionality IC. As a result, v′ and W in Equation (9)and Equation (11), are proportional to one another.

[0052] In step 17 of estimating pure spectrum and concentration,substituting Equation (11) obtained in step 15 into Equation (11)results in the following Equation (12):

A=F·S=F·W·IC  (12)

[0053] Consequently, referring to Equation (12), it is estimated thatthe product of the factor F and the mixing matrix W is the pure spectrumW of each component, and that the independent components C areproportional to the concentrations of the components contained in themixture.

[0054]FIG. 7 is a block diagram of an apparatus of estimating purespectra and concentrations of each component constituting a mixtureaccording to a preferred embodiment of the present invention.

[0055] Referring to FIG. 7, the apparatus includes a principal componentanalysis (PCA) unit 71, an independent component analysis (ICA) unit 73,a pure spectrum estimating unit 75 and a concentration estimating unit77.

[0056] The PCA unit 71 performs a principal component analysis of thespectra of the n mixtures, which are measured using m wavelengths, torepresent the spectra of the n mixtures as factors and scores of therespective factors. The ICA unit 73 performs an independent componentanalysis of the scores provided from the principal component analysisunit to decompose the scores into a mixing matrix and independentcomponents. The pure spectrum estimating unit 75 estimates the productof the factor provided from the principal component analysis unit andthe mixing matrix as the pure spectra of each component. Theconcentration estimating unit 77 estimates the independent components asthe concentrations of the components contained in the mixture.

[0057] Exemplary Embodiment

[0058] Twenty-five (25) sample water solutions, which were made bymixing glucose and sucrose with different concentrations, underwent PCAand then ICA. The pure spectrum of each of the glucose and sucrose isshown as FIG. 3. The spectra of the twenty-five (25) mixtures made bymixing the glucose and the sucrose with different concentrations areshown in FIG. 4.

[0059] The spectra of the twenty-five (25) mixtures as shown in FIG. 3are analyzed using the PCA and the ICA. Thus, the pure spectra of theglucose and sucrose are obtained as shown in graph G2 of FIGS. 5A and5B, respectively. Graph G2 is similar to graph G1, which shows theactual pure spectrum of each of the glucose and sucrose. In addition, asshown in FIGS. 6A and 6B, the independent components IC1 and IC2,obtained by applying the PCA and the ICA to the spectra of thetwenty-five (25) mixtures, are in a linear relationship with theconcentrations of the glucose and the sucrose.

[0060] In addition, the method of the present invention may be embodiedas a computer program on a computer-readable medium. For example, themethod of estimating the pure spectrum and the concentration of eachcomponent from n sample mixtures, in which p kinds of components aremixed to have different concentrations, may be embodied as a firstprogram for performing a principal component analysis (PCA) of thespectra of the n mixtures, measured using m wavelengths, to representthe spectra as factors and scores of the respective factors, and asecond program for performing an independent component analysis (ICA) ofthe scores produced by the first program to decompose the scores into amixing matrix and independent components, estimating the product of thefactor obtained by the first program and the mixing matrix as a purespectrum of each component, and estimating that the independentcomponents are proportional to the concentrations of the componentsmixed in the mixture.

[0061] The invention may be embodied in a general purpose digitalcomputer by running a program from a computer usable medium, includingbut not limited to storage media such as magnetic storage media (e.g.,ROMs, floppy discs hard discs, and the like), optically readable media(e.g., CD-ROMs, DVDs, and the like) and carrier waves (e.g.,transmissions over the Internet). The computer readable recording mediumcan be dispersively installed in a computer system connected to anetwork, and stored and executed as a computer readable code by adistributed computing environment. In addition, functional programs,codes, and code segments, required for embodying the present invention,may be easily inferred by programmers skilled in the art.

[0062] As explained so far, according to an embodiment of the presentinvention, as long as the principal component analysis (PCA) and theindependent component analysis (ICA) are applied to a spectrum of amixture, the number of components constituting the mixture and the purespectrum and the concentration of each component can be exactlyestimated. Moreover, the method of the present invention can be appliedto all types of spectral analyses, such as absorption spectrum,illumination spectrum, mass spectroscopy spectrum, magnetic resonancespectrum, and chromatography.

[0063] Preferred embodiments of the present invention have beendisclosed herein and, although specific terms are employed, they areused and are to be interpreted in a generic and descriptive sense onlyand not for purpose of limitation. Accordingly, it will be understood bythose of ordinary skill in the art that various changes in form anddetails may be made without departing from the spirit and scope of thepresent invention as set forth in the following claims.

What is claimed is:
 1. A method of estimating a pure spectrum and aconcentration of each component constituting n sample mixtures, in whichp kinds of components are mixed, the method comprising: (a) performing aprincipal component analysis of the spectra of the n mixtures, which aremeasured using m wavelengths, to represent the spectra of the n mixturesas factors and scores of the respective factors, wherein n, p, and m areintegers and m>p; and (b) performing an independent component analysisof the scores obtained in (a) to estimate the pure spectra and theconcentrations of the respective components.
 2. The method as claimed inclaim 1, wherein the number of factors to be used is decided from amongthe factors obtained in (a) and the independent component analysis isapplied to the scores of the decided factors.
 3. The method as claimedin claim 1, wherein the concentrations of the components constitutingthe mixture are statistically independent.
 4. The method as claimed inclaim 2, wherein the concentrations of the components constituting themixture are statistically independent.
 5. The method as claimed in claim1, wherein (b) comprises: (b1) performing the independent componentanalysis of the scores of the factors to decompose the scores into amixing matrix and independent components; (b2) estimating the product ofthe factors obtained in (a) and the mixing matrix obtained in (b1) asthe pure spectra of the respective components; and (b3) estimating theindependent components obtained in step (b1) as being proportional tothe concentrations of the components contained in the mixture.
 6. Themethod as claimed in claim 1, wherein (b) comprises: (b1) deciding thenumber of the factors to be used from among the factors obtained in (a);(b2) performing the independent component analysis of the scores of thedecided factors to decompose the scores into a mixing matrix andindependent components; (b3) estimating the product of the decidedfactor and the mixing matrix obtained in (b2) as the pure spectrum ofeach component; and (b4) estimating the independent components obtainedin (b2) as being proportional to the concentrations of the componentscontained in the mixture.
 7. The method as claimed in claim 1, whereinthe ICA may be performed using a technique selected from the groupconsisting of: maximization of non-gaussianity (MN), maximum likelihoodestimation (MLE), and minimization of mutual information (MMI).
 8. Themethod as claimed in claim 1, wherein in (a), a low-noise band includinginformation on a component to be estimated is decided as an analysisband from among the spectra of the n mixtures measured using mwavelengths.
 9. The method as claimed in claim 1, wherein (a) comprises:performing a preprocessing step to remove scattering and noise, beforeperforming the principal component analysis of the spectra of the nmixtures measured using m wavelengths.
 10. The method as claimed inclaim 9, wherein the preprocessing step comprises a technique selectedfrom the group consisting of multiplicative scatter correction (MSC),mean-centering, and autoscaling.
 11. A computer-readable medium havingembodied thereon: a first program for performing a principal componentanalysis of a spectra of the n mixtures measured using m wavelengths torepresent the spectra as factors and scores of the respective factors,wherein n and m are integers; and a second program for performing anindependent component analysis of the scores produced by the firstprogram to decompose the scores into a mixing matrix and independentcomponents, estimating that the product of the factor obtained by thefirst program and the mixing matrix is the pure spectra of eachcomponent, and estimating that the independent components areproportional to the concentrations of the components contained in themixture.
 12. The medium as claimed in claim 11, wherein theconcentrations of the components constituting the mixture arestatistically independent.
 13. An apparatus of estimating a purespectrum and a concentration of each component constituting n samplemixtures, in which p kinds of components are mixed, the apparatuscomprising: a principal component analysis unit for performing aprincipal component analysis of the spectra of the n mixtures, which aremeasured using m wavelengths, to represent the spectra of the n mixturesas factors and scores of the respective factors, where n, m, and p areintegers and m>p; an independent component analysis unit for performingan independent component analysis of the scores provided from theprincipal component analysis unit, to decompose the scores into a mixingmatrix and independent components; a pure spectrum estimating unit forestimating the product of the factor provided from the principalcomponent analysis unit and the mixing matrix as the pure spectra ofeach component; and a concentration estimating unit for estimating theindependent components as the concentrations of the components containedin the mixture.